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How to find highest power of 56 that 433! is divided
I've tried to use that power is equals to 433/56, but it is obviously not enough

Michael Rozenberg
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Student
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1 Answers1

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Calculate $$\left\lfloor\frac{433}{7}\right\rfloor+\left\lfloor\frac{433}{7^2}\right\rfloor+\left\lfloor\frac{433}{7^3}\right\rfloor.$$ I got $70$.

Thus, it's enough to check that $433!$ is divisible by $2^{3\cdot70}$, which is obvious because $$\left\lfloor\frac{433}{2}\right\rfloor+\left\lfloor\frac{433}{2^2}\right\rfloor+...>210.$$ Id est, $433!$ is divisible by $56^{70}$ and $70$ is a maximal power.

Deniz Tuna Yalçın
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Michael Rozenberg
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  • do I need also to divide 433 by powers of 8? – Student Oct 11 '17 at 08:45
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    @Student No, because $433!$ would naturally be divisible by $8$ more times as you also have factors of $4$ and $2$ contributing with factors of $8$ in $433!$. Technically you have to, but you realize that this will become higher when you take this into account. – skyking Oct 11 '17 at 08:47